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Almost Difference Sets and Reversible Divisible Difference Sets James A View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Richmond University of Richmond UR Scholarship Repository Math and Computer Science Faculty Publications Math and Computer Science 12-1992 Almost Difference Sets and Reversible Divisible Difference Sets James A. Davis University of Richmond, [email protected] Follow this and additional works at: http://scholarship.richmond.edu/mathcs-faculty-publications Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Davis, James A. "Almost Difference Sets and Reversible Divisible Difference Sets." Archiv Der Mathematik 59, no. 6 (December 1992): 595-602. doi: 10.1007/BF01194853. This Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has been accepted for inclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. Arch. Math., Vol. 59, 595-602 (1992) 0003-889X/92/5906-0595 $ 3.10/0 1992 Birkh/iuser Verlag, Basel Almost difference sets and reversible divisible difference sets By JAMES A. DAVIS 1. Introduction. Let G be a group of order mn and N a subgroup of G of order n. If D is a k-subset of G, then D is called a (m, n, k, 21,22) divisible difference set (DDS) provided that the differences dd' - i for d, d' e D, d 4= d' contain every nonidentity element of N exactly 21 times and every element of G - N exactly 22 times. Difference sets are used to generate designs, as described by [4] and [9]. D will be called an Almost Difference set (ADS) if )ol and 22 differ by 1. The reason why these are interesting involves their relationship to symmetric difference sets. A symmetric difference set is a DDS with 21 = 22, so the ADS are "almost" difference sets. Symmetric difference sets are becoming more and more difficult to construct, and this is as close as we can get with a divisible difference set. One helpful way to view DDS is to consider the group ring Z[G]. We will abuse notation by associating to the subset A of G the element A = ~ a of the group ring. a~A Similarly, we will write A (- l) = ~ a- i. The definition of a DDS immediately yields the aeA group ring equation DD (-1) = k + 2i(N - 1) + ,~2(G -- N). This is often a very simple way to check whether a subset of a group meets the properties of being a divisible difference set, and it has the advantage that it works in nonabelian cases. We will use this technique to check the second construction. We now restrict our attention to abelian groups; in this case, characters of the group are simply homomorphisms from the group to the complex numbers (we are following the lead of Turyn [13]). Extending this homomorphism to the entire group ring yields a map from the group ring to a number field. The character sum for the character Z on the element D of the group ring yields 3 possible results: Z(D) = k if )f is the principal (all l) character, Iz(D)l =~,/~-2i if Z is nonprincipal on N, and Iz(D)I= xf k - )u + (21 - '~2) n if)~ is principal on N but nonprincipal on G. One other useful fact about characters is that if Z is a nonprincipal character on a group (or subgroup) H, then the character sum ofz over H will be 0. If we have a subset of the group that satisfies these character sums for every character Z, then that subset will be a DDS (this is because of the orthogonality relations for characters: see [13] for similar arguments). Thus, our strategy will be to come up with a "candidate" subset of the group, then use character theory to check that all the sums are correct. We will use this techniques to check the first construction. 38* 596 J.A. DAWS ARCH. ~a.TH. One other condition that we wilt note in this paper involves the multiplier group of the difference set. We say that a difference set is reversible ifD (- ~) = D. This is a nice property for a set to have because it has strong implications for all of the other multipliers. For symmetric difference sets, it is conjectured that there is only one non-Menon difference set with multiplier - 1. Reversible DDSs have been the subject of much recent research, with both constructions and nonexistence results appearing in the literature: In particu- lar, Ma [!0, 1t] ha s provided several nonexistence results based on number theoretic considerations on the parameters. Jungnickel [7, 8], and then Arasu, Jungnicket, and Pott [2] have provided constructions of reversible DDSs based on the constructions for re- versible symmetric difference sets found in McFarland's paper [10]. Construction 2 will yield some new reversible DDSs that generalize the constructions found in these papers. 2. Construction 1, The first construction has connections to a familiar set of parameters from symmetric difference sets. Turyn [14] has constructed (4.32~ 2.320 -3~, 3 ~" - 3") in groups of the form H x EA (32~ where H is either group of order 4, and EA (32,) is the elementary abelian group of order 32". In an effort to find other groups that have difference sets with these parameters, this author constructed divisible difference sets; but these divisible difference sets did not have 2~ close to 22 (see [5]). Since then, this author, together with Jedwab, Arasu, and Sehgal have constructed difference sets with the same parameters as Turyn's examples (see [1]). A minor adjustment of that difference, set will yield an ADS. Let G be the group of the form H x Zzo where H is either group of order 4. The generators of the 3 a parts are y and z, and we will write the elements of H as ho, hi, h 2, h 3. The subgroup N is the group (y, z> ~ Z~o. We also want to label the cyclic subgroups 3 '+~ - 1 of order 3 ~ in a careful way. We wilt use D1, ~ = (yz~>, i = 0, 1, ..., 3 -- i - 2 and 3 ~ - I D i,1 = <y3jz),j = 0, 1 ..... 3-~-i-- 1. It is worth noticing that O~,,. = D~+3. and D., ~ = D, + 3-- ~, t. Consider the set 3 ~ -- 1 3 ~ -- t ........ ! ....... t D= k=o hk i=o~ ziD1,31+k W 3 j~o yiDj'I " This is the "candidate" subset; we claim that this is an ADS with the parameters (4,32~ 2(3 z~ - 3"), 320 - 2.3", 32" - 2.3 ~ + 1). The only difference between this and the difference sets found in [1] is this construction is missing one coset of the D~,j. The proof is broken down into the following sequence of lemmas (these are the essentialIy the same as found in [1], and we do not include the proofs here). I_emma 2.1. D has no repeated elements. This ensures us that we wilt have the correct number of elements in the difference set, and that there is no overlapping of the cosets. Lemma 2.2. If Z is a character of order 3" on G/H, then Z is nonprincipaI on all of the Di, j except one, where it is principal. Vol. 59, 1992 Almost difference sets 597 This lemma declares that any character of order 3" will have a character sum of 0 over all of the Di, j except one. Thus, the character sum over the entire set will be (in modulus) the size of that D~,j, which is 3". The only difficulty in the proof involves cases when the D~,j shows up twice. In that case, the character sum will be 3"7~(z~) (1 + ~3o-1), where ~ is a primitive 3" root of unity. The modulus of that sum is 3 ~ so it matches what is required. The next lemma describes the situation for all of the other characters that are nonprin- cipal on the forbidden subgroup (y, z). Lemma 2.3. If )~ is a character of G/H that is nonprincipal but of order less than 3", then 3 a - 1 3 a - 1 x(ziD1,3i+k) = 2 7~(YJDj,O =0 i=O 1=0 for every k except I, where the character sum is 3" in modulus. Thus, the character sum is correct for all of the characters that are nonprincipal on (y, z). All that remains is to check the character sums for the characters that are principal 32~ __ 3a on (y, z), but nonprincipal on H. Each of the cosets of (y, z) has -- elements, so 32" - 3" 3 2 the character sum is --2 ~=o~ z(hl) = 0 = ,fk - 21 + (2 z - 22) n, which is correct. The next lemma summarizes that result. Lemma 2.4. If Z is a nonprincipal character on G that is principal on (y, z), then )~(D) = O. Putting all this together, we get the following Theorem 2.1. D is a (4,32a, 2(3 2a -- 3~), 32" - 2 3", 3 2a -- 2- 3 a + 1) ADS in G. This DDS generates a divisible design with the same parameters, and that design is semi-regular since k 2 = (2(32" - 3a)) 2 -= 4' 32"(3 a - 1) 2 --= mn2z. The author has not found this set of parameters in the literature, so this seems to be a new set of parameters for a semi-regular design that is also an ADS.
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